Quaternions $\Leftrightarrow $ Rotations -Conceptual question

Question

I have an angular velocity vector analogue defined as(called Darbaoux vector some times in curve)

$ \omega(s)=\frac{\mathrm{d}\theta(s) }{\mathrm{d} s}=\delta\theta(s)_x \vec{i}+\delta\theta(s)_y \vec{j}+\delta\theta(s)_z \vec{k} \tag 1$

And we have a moving material frame on curve as shown in figure

.

Specifications

1. The material frame rotates around angular velocity vector $\omega(s)$ at each point an angle $\delta\theta(s)$ .The magnitude of angular velocity vector gives the small angle of rotation it makes that is $|\delta\theta(s)|$
2. It implies the axis of rotation for each matrial frame at s is $\omega(s)$ and it makes small rotations $|\delta\theta(s)|$ at each s while travelling therough the curve
3. We can make a rotation 3D Matrix $R(s)_{3 \times 3}$ by arranging the orthogonal directors $d_3(s),d_2(s),d_1(s)$ in column order. Means place $d_3(s)$ as column $1$,$d_2(s)$ as column $2$,$d_1(s)$ as column $3$

Question

1. If we are only given the function $\omega(s)$ can we find out $R(s)$?

NB: I have one solution in my mind. I am writing that as my attempt below please verify it. Thanks

Answer 1

ATTEMPT #1

Preliminary Supporting data for the solution

1. Every rotation matrix has an angle of rotation$(\alpha)$ and unit axis of rotation$(\hat{a})$ has the quaternion form

$Q=cos(\frac{\alpha}{2})+\hat{a}\hspace{.2cm} sin(\frac {\alpha}{2}) \tag 2$

2. Let $Q=(w,l,m,n )=(\hat{v },w) $ represents quaternion and unit $\hat{a}=(L,M,N)$

$ \begin{eqnarray} l=L sin(\frac {\alpha}{2})\\ m=M sin(\frac {\alpha}{2})\\ n=N sin(\frac {\alpha}{2})\\ w= cos(\frac{\alpha}{2}) \end{eqnarray} \tag 3$

3. 3D Rotation Matrix representation of Q can be written as

$ \begin{bmatrix} 1 - 2m ^2 - 2n ^2&2 l m + 2 w n & 2 l n - 2 w m \\ 2 l m - 2 w n &1 - 2 l ^2 - 2 n ^2 &2 m n + 2 w l \\ 2 l n + 2 w m & 2 m n - 2 w l & 1 - 2 l ^2 - 2 m ^2\end{bmatrix}\tag 4$ Solution

1. Let $\alpha = |\omega(s) |$ and axis of rotation $\hat{a}=\frac{\omega(s) }{|\omega(s) |}$.It is mentioned already because frame rotates around $\omega(s)$
2. Find out Q for it and then convert that to 3D rotation matrix as in equation (4). What you get is R(s).

Please comment and inform me whether I am correct or not